Problem: The image of the point with coordinates $(1,1)$ under the reflection across the line $y=mx+b$ is the point with coordinates $(9,5)$.  Find $m+b$.
Answer: The line of reflection is the perpendicular bisector of the segment connecting the point with its image under the reflection.  The slope of the segment is $\frac{5-1}{9-1}=\frac{1}{2}$.  Since the line of reflection is perpendicular, its slope, $m$, equals $-2$.  By the midpoint formula, the coordinates of the midpoint of the segment are $\left(\frac{9+1}2,\frac{5+1}2\right)=(5,3)$.   Since the line of reflection goes through this point, we have $3=(-2)(5)+b$, and so $b=13$.  Thus $m+b=-2+13=\boxed{11}.$